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Theorem rexlimddvcbv 40832
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40830. The equivalent of this theorem without the bound variable change is rexlimddv 3283. Usage of this theorem is discouraged because it depends on ax-13 2392, see rexlimddvcbvw 40831 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
rexlimddvcbv.1 (𝜑 → ∃𝑥𝐴 𝜃)
rexlimddvcbv.2 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
rexlimddvcbv.3 (𝑥 = 𝑦 → (𝜃𝜒))
Assertion
Ref Expression
rexlimddvcbv (𝜑𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv
StepHypRef Expression
1 rexlimddvcbv.1 . 2 (𝜑 → ∃𝑥𝐴 𝜃)
2 rexlimddvcbv.3 . . 3 (𝑥 = 𝑦 → (𝜃𝜒))
3 rexlimddvcbv.2 . . 3 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
42, 3rexlimdvaacbv 40830 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
51, 4mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139
This theorem is referenced by: (None)
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